Rotation Studies in Fusion Plasmas via Imaging X-ray Crystal Spectroscopy

PSFC/RR-08-7

DOE/ET-54512-362

Rotation Studies in Fusion Plasmas via
Imaging X-ray Crystal Spectroscopy

Alexander Charles Ince-Cushman

September 2008
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge MA 02139 USA

This work was supported by the U.S. Department of Energy via Co-Operative Agreement
No. DE-FC02-99ER54512-CMOD. Reproduction, translation, publication, use and
disposal, in whole or in part, by or for the United States government is permitted.


Rotation Studies in Fusion Plasmas via Imaging
X-ray Crystal Spectroscopy
MASSACHUSETTS INSTiTUTE
OF TECHNOLOY

by

Alexander Charles Ince-Cushman
B.A.Sc., Aerospace Engineering,
University of Toronto (2003)

AUG 19 2009

L1E~t~RtRE3

Submitted to the Department of Nuclear Science & Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2008

@ Massachusetts Institute of Technology 2008. All rights reserved.
ARCHIVES
Author ...........
Department of Nuclear Science & Engineering
August 28, 2008
A
Certified by.........

Si

Dr. John E. Rice
Principal Research Scientist
Thesis Supervisor
/ , L_

Certified by.............
Prof. Ian H. Hutchinson
Nuclear Science & Engineeri g Department Head
Th s Reader
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Accepted by...
"' Prof' acquelyn C. Yanch
Professor of Nuclear Science & Engineering
Chair, Department Committee on Graduate Students



Rotation Studies in Fusion Plasmas via Imaging X-ray
Crystal Spectroscopy
by
Alexander Charles Ince-Cushman
Submitted to the Department of Nuclear Science & Engineering
on August 28, 2008, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy

Abstract
The increase in plasma performance associated with turbulence suppression via flow
shear in magnetically confined fusion plasmas has been well documented. Currently,
the standard methods for both generating and measuring plasma rotation involves
neutral beam injection (NBI). In the large, high density plasmas envisaged for next
generation reactors, such as ITER, NBI will be considerably more difficult than in
current experiments. As a result, there is a need to identify alternative methods for
generating and measuring plasma flows. In an effort to meet these needs, a high resolution x-ray crystal spectrometer capable of making spatially resolved measurements
has been designed, built, installed and operated on the Alcator C-Mod tokamak. By
taking advantage of toroidal symmetry and magnetic flux surface mapping it is possible to perform spectral tomography with a single fan of views. This combination of
spatially resolved spectra and tomographic techniques has allowed for local measurement of a number of plasma parameters from line integrated x-ray spectra for the first
time. In particular these techniques have been used to measure temporally evolving
profiles of emissivity, charge state densities, rotation velocities, electron temperature,
ion temperature, as well as radial electric field over most of the plasma cross section

(r/a < 0.9). In this thesis three methods for the generation of flows without the use
of NBI are identified; intrinsic rotation in enhanced confinement modes, lower hybrid
wave induced rotation and ICRF mode conversion flow drive. Each of these methods
is discussed in detail with reference to how they might be used in next generation
tokamaks.
Thesis Supervisor: Dr. John E. Rice
Title: Principal Research Scientist
Thesis Supervisor: Prof. Ian H. Hutchinson
Title: Nuclear Science & Engineering Department Head


Acknowledgments
There are a great many people who have helped to produce the work contained in the
pages that follow. Without their collective efforts this thesis would simply not have
been possible. First and foremost I would like to thank Dr. John Rice for his uncanny
ability to blend perfectly the roles of adviser, teacher and friend. He has taught me
virtually everything I know about the art and science of being an experimentalist for
which I am eternally grateful. There is a number of ways in which John has helped
me throughout the years I have known him, not least of which is his unwavering
dedication to grammatical correctness (Split infinitives beware, dangling modifiers be
gone, and let the data speak for themselves).
My fellow graduate student Mathew Reinke deserves special praise for the tremendous contributions he has made to this work. Among other things, he is responsible
for the development and implementation of a spectral tomographic algorithm that
was crucial for this thesis. The thoughtfulness and attention to detail that he brings
to his work is extraordinary. I can not thank him enough for all his help.
I would also like to acknowledge my collaborators from the Princeton Plasma
Physics Laboratory; Doctors Manfred Bitter, Kenneth Hill and Steve Scott. It was
Dr Bitter who first conceived of the idea of an imaging x-ray crystal spectrometer.
Moreover, it was the early experiments of Dr. Bitter and Dr. Hill on this type of
instrument that laid the foundation for the spectrometer that is the focus of this

thesis. Their helpful guidance through the design, construction and operation of
the spectrometer was invaluable. I would like to thank Dr. Earl Marmar, for his
encouragement and for ensuring that this project had all the resources needed for
success.
I would like to thank Ming Feng Gu for providing the atomic physics data used
throughout the thesis. Here too, Mathew Reinke, deserves the credit for using these
data to calculate electron temperature and charge state density profiles based on
line emission spectra. I would like to acknowledge the members of the Alcator CMod Lower Hybrid Current Drive group (Prof. Ronald Parker, Gregory Wallace, Dr.


Shunichi Shiraiwa, Dr. Randy Wilson, and Orso Meneghini) for their collaborations
on joint rotation/lower hybrid experiments. I would like to thank Dr. Yijun Lin for
spearheading the investigations of mode conversion flow drive experiments on Alcator
C-Mod and his patience in teaching me about this fascinating field of study.
A number of people generously provided data were was used in this thesis. They
are: Amanada Hubbard (electron temperature from electron cyclotron emission),
Jerry Hughes (electron temperature and density from Thomson scattering), Steve
Wolfe (magnetic equilibrium calculations), Catherine Fiore (central ion temperature
from neutron emission), Rachael McDermott (ion temperature and radial electric field
from charge exchange spectroscopy) and Jim Irby (electron density from interferometry).
I would like to thank my thesis reader, professor Ian Hutchinson for his careful reading of this document and a number of fruitful discussions that undoubtedly
improved the final result. I would like acknowledge the other members of my thesis committee, professors Jeffrey Friedberg and Dennis Whyte. In addition to those
mentioned above I would like to thank all of the scientists, technicians, professors, administrators and engineers that make up the Alcator C-Mod community, all of whom
have helped me over the years: Abhay Ram, Andy Pfeifer, Bob Childs, Bob Granetz,
Brain LaBombard, Bruce Lipschultz, Clare Egan, Corrine Fogg, Darin Ernst, Dave
Arsenault, Dave Belloffato, Don Nelson, Dorian McNamara, Dragana Zubcevic, Earl
Marmar, Ed Fitzgerald, Felix Kreisel, Gary Dekow, Heather Geddry, Henry Bergler,
Henry Savelli, James Zaks, Jason Thomas, Jessica Coco, Jim Terry, Joe Bosco, Joe
Snipes, Josh Stillerman, Lee Keating, Leslie West, Liz Parmelee, Marcia Tench-Mora,
Mark Iverson, Mark London, Martin Greenwald, Matt Fulton, Michael Rowell, Miklos Porkolab, Patrick MacGibbon, Paul Bonoli, Paul Lienard, Paul Rivenberg, Peter

Brenton, Peter Catto, Peter Koert, Rachel Morton, Ravi Gondhalekar, Richard Murray, Rick Leccacorvi, Rosalie West, Rui Vieira, Steve Wukitch, Ted Biewer, Tom Fredian, Tommy Toland, Valerie Censabella, William Burke, William Byford, William
Parkin, and Xiwen Zhong.
I would also like to thank my fellow graduate students for their help, camaraderie,


and control room antics over the years: Aaron Bader, Alexandre Parisot, Andrea
Schmidt, Arturo Dominguez, Brock Bose, Eric Edlund, Gregory Wallace, Igor Bespamyatnov, Jason Sears, Jinseok Ko, John Liptac, Kelly Smith, Kenneth Marr,
Laurence Lyons, Liang Lin, Marco Ferrara, Matthew Reinke, Nathan Howard, Noah
Smick, Rachael McDermott, Scott Mahar, and Vincent Tang.
Finally, I would like to thank my parents Sue and Paul, my brother Daniel and
the entire MacDonald clan who collectively make up my family in the truest sense of
the word.


Contents

1 Introduction

17

1.1

Outline ..................

...............

1.2

The Alcator C-Mod Tokamak ...................

1.3

Units .................

18
...

19

..................

20

2 Imaging X-ray Crystal Spectrometers
2.1

Bragg Reflection

2.2

Johann Spectrometers

23

..................

.........

...................

.

.......

24
25

2.2.1

Spherically Bent Crystal Optics . ................

2.2.2

Spatial Resolution

...................

.....

28

2.2.3

Detector Alignment ...................

.....

29

2.2.4

Johann Error and Spectral Resolution

27

. ............

.......

30

2.3

Emission Line Selection .......

.........

2.4

Crystal Selection ...................

.........

34

2.5

Chapter Summary

.........

34

...................

32

3 The Spatially Resolving High Resolution X-ray Spectrometer: HirexSr

37

3.1

Design Criteria ...................

3.2

Design Constraints ...........

3.3

Component Descriptions ...................
3.3.1

Crystals ...................

3.3.2

X-ray Detectors ...................

. .
..

......
.........

37
38

......
.

.

.........
.......

40
40
40


3.4

3.3.3

Detector Mounting .......

3.3.4

Alignment Stages ........

3.3.5

Base Plate & Housing.....

3.3.6

Spectrometer-Reactor Interface

Chapter Summary

...........

4 Inferring Plasma Parameters from Line Emission Radiation
4.1

Doppler Shifts . . .

4.2

Line Ratio Measurements

4.3

Data Analysis .

4.4

4.5

..........................

49

....................

51

..........................

52

4.3.1

Wavelength Calibrations ......

4.3.2

Multi-line Fitting .

4.3.3

Spectral Tomography ...................

Example Profiles .

...........

....................

56
57

........................

60

4.4.1

Emissivity Profiles .

4.4.2

Charge State Density Profiles .

4.4.3

Electron Temperature Profiles ..............

4.4.4

Toroidal Rotation Profiles .

4.4.5

Ion Temperature Profiles ......

4.4.6

Radial Electric Field Profiles .

Chapter Summary

. .

...................
.....

.....

.....

.......................

5 Rotation Theory

52

60
........

64
64

..........

67

...........

69

.........

71
77
79

5.1

Neoclassical Rotation Theory

79

5.2

Sub-Neoclassical Theory . . ..

82

5.3

ICRF Induced Rotation

. . ..

84

5.4

Accretion Theory . . . . . . ..

84

5.5

Flow Drive via Reynolds Stress

86

5.6

Summary of Rotation Theories

88


91

6 Intrinsic Rotation in Enhanced Confinement Regime Plasmas
6.1

96

Multi-Machine Intrinsic Rotation Database ................

6.2 Chapter Summary

104

.......................

.....

105

7 Lower Hybrid Wave Induced Rotation Profile Modification

106

7.1

Temporal Evolution ............................

7.2

LH Induced Rotation Modifications and Normalized Internal Inductancel06

7.3

LH Induced Rotation Modification in H-mode Plasmas . .......

109

7.4

Spatial Extent of Rotation Modifications . ...............

111

7.5

Lower Hybrid Induced Fast Electron Pinch ... . ...........

111

7.6

Chapter Summary

114

.........

...................

115

8 ICRF Mode Conversion Flow Drive
8.1

...........

Toroidal Rotation ...............

. . . .

8.2 Poloidal Rotation and Radial Electric Field
8.3 Comparison with Theory . . . . . . . ....

8.4 Direct ICRF Momentum Input
8.5 Chapter Summary

. . . . . . .

..............

. 116
. . . . .

...........

.

125

...........

.

127

...........

. 129
131

9 Conclusions and Future Work
9.1

Imaging X-ray Crystal Spectroscopy .

9.2 Intrinsic Rotation Studies

. 119

. . . . . . 131

............

. . . . . . 133

..................

9.2.1

Intrinsic Rotation in Enhanced Confinement Modes . . . . . . 133

9.2.2

Lower Hybrid Wave Induced Rotation

9.2.3

ICRF Mode Conversion Flow Drive . . . . . . ...

9.3 Conclusions . .

. . . . . . . . . . . . . 134

. . . . . . 135

.........................

137

A H- & He-Like Argon Spectra
A.1 He-like Argon Spectra .
A.2 H-like Argon Spectra .....

. . . . . . 135

138

........................
. .. . .

. .

...............

139

B X-ray Transmission Coefficients and Helium Purity Measurements 141


C Interspecies Thermal Equilibration

147

C.1 Ion-Impurity Thermal Equilibration . . . . . . . .

148

C.2 Electron Ion Thermal Equilibration . . . . . . ..

149

D Resolving Power and Instrumental Temperature

155

E Ion-Impurity Parallel Flow Separation

161

F Intrinsic Rotation Scaling Variable Description

163

F.1

Derivation of mave

. . . . . .

164

F.2 Derivation of the Ion Acoustic Sound Speed . . .

165

G Rotation Generation Through Momentum Diffusivity Asymmetries167
G.O.1

Symmetry Breaking .........

........

.......

171


List of Figures
1-1

. . . .

The Alcator C-Mod Tokamak . . . . . . ...

2-1 Bragg Reflection

. 20

. . . . 25

...............

2-2 A Simple Flat Crystal Spectrometer . . . . .

. . . . 26

2-3 Rocking Curve

. . . . 27

................

2-4 Single Bent Johann Crystal Spectrometer ..

. . . . 28

2-5 Spherically Bent Crystal Optics . . . . . . .

. . . . 29

2-6 Spatial Resolution of a Johann Spectrometer

. . . . 30

2-7 Detector Alignment ..............

. . . . 31

2-8 Johann Error .................

. . . . 32

2-9 Coronal Equilibrium For Noble Gases . . . .

. . . . 33

3-1 Spectrometer Layout .......
3-2 Pilatus 100k X-ray Detector .
3-3 Modified Detector Arrangement .
3-4 He-like Detector Array ......
3-5 Top Down View of Spectrometer
3-6 Isometric Spectrometer View .
3-7 Beryllium Window

........

3-8 Spectrometer/Reactor Interface .
4-1 Raw Image of the He-like Argon Spectra
4-2 Curvature Correction Residual ......
4-3

Curvature Correction

. . . . . . . ..

. . .

. . . .

.

55

4-4

Multi-line Fitting .............................

4-5 Emissivity Contour Plot of the w Line from the He-like argon spectra.
4-6

Emissivity Surface Plot of the w Line From the He-like Argon Spectra.

4-7 Up/Down Asymmetry Emissivity Comparison
4-8

Charge State Densities Profiles

.....

63
..........

..

4-9 Line Ratios From He-like Ar Spectra for Te Measurements
4-10 Electron Temperature as a Function of Line Ratios

65
66
67

........

. .................

4-13 Toroidal Rotation Frequency Surface Plot

......

. .........

4-11 Electron Temperature Profiles From Emissivity Ratios
4-12 Toroidal Rotation Profiles

. . .

68

. .. . ...
. ..............

69
70

4-14 Ion Temperature Profile Evolution . ................

. .

72

4-15 Ion Temperature Profile Surface Plot . .................

73

4-16 Impurity and Electron Temperature Profiles . .............

74

4-17 Comparison of Ti Based on Line Averaged and Inverted data.
4-18 Radial Electric Field Profile ...............

.....

75

. . .....

76

4-19 Comparison of Radial Electric Field Measurements With CXRS . . .

77

4-20 Er Profile evolution Through An L-H transition . . . . .

. . .

.

78

6-1

Definitions of AW and AV

. . .

.

92

6-2

The Rice Scaling . .

. . .

.

93

6-3

Toroidal Rotation Evolution Through an L-H Transition . . . . . . . 94

6-4

The Rice Scaling For C-Mod, DIII-D and Tore-Supra . . . . . . . . . 95

6-5

Ion Thermal Mach Number vs.

6-6

Alfvnic Mach Number vs.

................

.....................

3

N

............

N ..................

. . .

. 98

. . .

. 98

6-7 Ion Thermal Mach Number vs. Normalized Gyro-radius . . . . . . . .

99

6-8

Alfvnic Mach Number vs. Normalized Gyro-radius . . . . . . . . . . 100

6-9

Ion Thermal Mach Number vs. Collisionality . . . . . . .

. . . .

. 100

6-10 Alfvnic Mach Number vs. Collisionality .........
6-11 Dimensionless Scaling of Intrinsic Rotation for Alfvnic Mach Number
6-12 Machine Parameter Intrinsic Rotation Scaling of AV

102

. . . . . . . . . 103


7-1

7-2 Lower Hybrid Phase Scan
7-3

107

Lower Hybrid Induced Rotation Time Histories . ...........

AV vs. Ali ......

108

.....

...................

109

.............

.........

110

7-4 LH Rotation Modification in H-mode . .................
.

112

7-5

Spatial Profiles Associated With Lower Hybrid Induced Rotation

7-6

E, Profile Evolution With and Without LHCD .............

8-1

Comparison of Mode Conversion and Minority Heating ICRF Discharges117

8-2

Comparison of the Rice Scaling for Discharges With MC and MH ICRF. 118

8-3

Radial Profiles of Toroidal Rotation and Ion Temperature in Discharges

113

8-4

120

...

With and Without MC ICRF. ...................

Temporal Evolution of the Toroidal Rotation in a Discharge with MC
ICRF. ..................

.......

121

. ........

122

.........

8-5

Mode Conversion Induced Poloidal Rotation Profiles

8-6

Radial Electric Field Profiles in MH and MC plasmas .........

123

8-7 Central Toroidal Rotation vs. Time For An MC ICRF Discharge ...

129

A-1 The He-like Argon Spectrum as Measured by the Hirex-Sr Spectrometer138
A-2 The H-like Argon Spectrum as Measured by the Hirex-Sr Spectrometer 139
..

A-3 Raw image of the H-like Spectra ...................

140

B-1 3.1 keV X-ray Transmission Through a Mixture of Helium and Air . . 143

B-2 Percent Air Impurity and Transmission Coefficient Evolution .....

144

C-1 Equilibration Times in a High Density/Low Temperature Discharge . 150
C-2 Comparison of Ti and Te in a High Density/Low Temperature Discharge151
C-3 Comparison of Ti and Te in a High Temperature Discharge ......

152

C-4 Equilibration Times in a High Temperature Discharge .........

153

D-1 Multi line fit of an H-like spectrum using Voigt functions .......

159

E-1 AVll,

162

in a typical Alcator C-Mod discharge

. .............

G-1 Thermal Momentum Density Profiles . .................

170


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List of Tables
1.1

. . . . . 21

Alcator C-Mod Facility Parameters ..............

. . . . . 40

.....................

3.1 Crystal Specifications . .
3.2 Detector Specifications . .

. . . . . 42

....................

. . . . . 45

.......................

3.3 Alignment Stages . .
5.1

Neoclassical Asymptotic Dimensionless Viscosity Coefficients

82

6.1

Intrinsic Rotation Database Parameter Ranges . . . . . . . ..

96

8.1

MC Plasma Parameters

.

127

....................

A.1 Argon lines in the Wavelength Range 3.94A < A < 4.00A . .

138

. . .

139

A.3 Molybdenum Lines in the Wavelength Range 3.72 < A < 3.80

140

A.2 Argon Lines in the Wavelength Range 3.72 < A < 3.80

. . . . . . . . . .

163

F.2 Geometric Quantities . . . . . .

. . . . . . . . . .

163

F.3 MHD Variables . . . . . . ...

. . . . . . . . . .

164

F.4 MHD Variables . . . . . . ...

. . . . . . . . . .

164

F.5 Velocities

. . . . . . . . . .

165

. . . . . . . . . .

165

F.1 Plasma Quantities

.......

............

F.6 Mach Numbers . . . . . . ...


THIS PAGE INTENTIONALLY LEFT BLANK


Chapter 1
Introduction
One of the fundamental problems facing the magnetic confinement fusion community
is the reduction in performance associated with plasma turbulence. In present experiments, turbulence is the limiting factor on temperature and density gradients which
in turn limit the rate of fusion energy production. Although much remains to be done,
progress has been made both theoretically and experimentally on understanding and
characterizing this turbulence.
An example of this improved understanding is the recognition of the importance
of rotation on plasma performance. Strong plasma rotation can help stabilize destructive magneto-hydrodynamic instabilities (resistive wall modes, RWMs [1], [2])
while gradients in rotation can improve confinement by suppressing turbulence[3],[4].
In many experiments the rotation profiles associated with improved performance are
generated through the use of neutral beam injection (NBI) [5]. Unfortunately this approach may prove impractical in the large, high density plasmas envisioned for next
generation devices such as ITER [6], [7]. As a result, there is a need to develop alternatives to NBI for driving strong plasma rotation. Significant self generated flows
have been observed on a number of tokamaks [8] suggesting that it may be possible
to reap the benefits of rotation without the use of NBI. Furthermore, chapters 7 and
8 of this thesis discuss recent observations of significant flow drive by wave heating.

While these alternative approaches to flow generation have been receiving increased study in recent years, they remain poorly understood theoretically and sparsely


measured experimentally. This paucity of measurement can be explained by the fact
that the standard method for measuring rotation profiles is active charge exchange
spectroscopy with NBI. The neutral beams involved generally represent large sources
of external momentum making it very difficult to study intrinsic flows. Passive x-ray
spectroscopy has been used to diagnose intrinsic rotation without the use of perturbative beams but is often limited to a just of few line-integrated measurements.
To address the need for rotation profile measurements without the use of neutral
beams, a new type of x-ray crystal spectrometer has been developed. This instrument
combines the passive nature of standard x-ray spectroscopy with the ability to make
spatial resolved measurements. The contents of this thesis are divided between two
broad themes. The first involves the theory and design of imaging x-ray crystal
spectrometers. The second deals with plasma flows experiments made possible by
the expanded diagnostic capabilities of the new instrument.

1.1

Outline

This thesis is divided into 9 chapters and 7 appendices. The topic of imaging x-ray
crystal spectrometers is addressed in chapters 2, 3 and 4. Specifically, chapter 2
summarizes the physics of such an instrument and how to go about designing one.
Chapter 3 describes the imaging x-ray crystal spectrometer that was designed and
built to obtain the data for the second half of this thesis. Chapter 4 describes how
plasma parameters can be inferred from the line emission spectra the aforementioned
spectrometer was designed to measure.
Chapters 5 through 8 focus on the study of tokamak plasma flows in the absence
of external momentum input from neutral beams. Chapter 5 provides an overview of
rotation theories and discusses them in the context of rotation measurements made to

date. Chapter 6 describes intrinsic rotation in enhanced confinement regime plasmas.
Chapters 7 and 8 cover recent measurements of rotation drive by lower hybrid waves
and mode converted ICRF1 waves, respectively. The final chapter, Chapter 9, briefly
summarizes the key findings of the thesis and outlines some possible directions for
'Ion cyclotron range of frequency


future work. Appendix A provides details on the line emission spectra of hydrogenand helium- like argon. Appendix B outlines the calculation of the x-ray transmission
through the helium-air mixture present in the spectrometer housing. Appendix C
consists of calculations of interspecies thermal equilibration times. This appendix
also provides evidence of the strong thermal coupling between impurity and bulk ions
in Alcator C-Mod plasmas and discusses the conditions under which temperature
separation between ions and electrons is to be expected. Appendix D covers the
concept of instrumental temperature and describes a novel technique for determining
the resolving power of a spectrometer. Appendix E discusses the separation of ion and
impurity parallel flows in tokamak plasmas. Appendix F provides a summary of the
variables used in the multi-machine intrinsic rotation database as well as derivations
of approximate expressions for the average ion mass, and the acoustic sounds speed.
Appendix G discusses how toroidal rotation can be generated through asymmetries
in momentum diffusivity.

1.2

The Alcator C-Mod Tokamak

The spectrometer that is the focus of the first part of this thesis was installed on
the Alcator C-Mod tokamak at the MIT Plasma Science and Fusion Center. Alcator
C-Mod is a compact tokamak designed to operate at the high magnetic field and
densities envisaged for burning plasmas on reactors such as ITER and DEMO. Figure
1.1 shows isometric and cross sectional views of the device while table 1.1 summarizes

its key parameters.
Alcator C-Mod is the ideal device for conducting research on the twin goals of this
thesis, namely the development of high resolution imaging x-ray spectroscopy and
intrinsic rotation studies. High resolution x-ray spectroscopy is made easier by the
enormous amount of line radiation emitted by the high density discharges produced
on Alcator C-Mod 2. Furthermore Alcator C-Mod does not use NBI for heating which
allows for the study of intrinsic plasma rotation on every discharge.
2

For a fixed impurity fraction, radiated power is proportional to the square of electron density


-'-

~-:::::;:
'-

-::~0

Figure 1-1: The Alcator C-Mod tokamak

1.3

Units

The units used throughout this thesis are in the system international, SI, and specifically MKS. The one exception is that temperatures are quoted in electron volts (where
leV - 11,600 'K) since writing and reading sentences like "the plasma was heated to
60,000,000 oK" is somewhat cumbersome.

Table 1.1: Alcator C-Mod Facility Parameters
Parameter
Major Radius
Minor Radius
Plasma Volume

Value/Range
R - 0.68 m
a ~ 0.22 m
V lm 3

Plasma Surface Area

S

Maximum Toroidal Field
Maximum Plasma Current
Elongation
Triangularity

BT < 8 T
I, < 2 MA
e < 1.9
6 < 0.85

Maximum Toroidal Field Pulse length

5 s >>

Ion Cyclotron RF source Power

Lower Hybrid RF Source Power
Collisionality range

8 MW, 50 to 80 MHz
3 MW, 4.6 GHz
0.05 < v* < 10

Normalized pressure

/N < 1.8

Absolute Plasma Pressure

< 0.2 MPa (volume average)

-

7m 2

TCR


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Chapter 2
Imaging X-ray Crystal
Spectrometers
High resolution measurements of line radiation from partially ionized atoms have
been used to diagnose fusion plasmas for a number of years[9]. While this method

has been successfully employed on a variety of experiments, its usefulness has been
limited by the lack of spatial localization associated with the line integrated nature
of the measurement.
The problem of spatial localization can be overcome if spectra from multiple lines
of sight are available. The general problem of inferring local quantities from a set
of line integrated measurements is not unique to plasma physics. Indeed, the entire
discipline of tomography was developed to address it. The analytic methods required
for performing spectro-spatial tomographic inversions of line emission spectra have
been known for some time ([10], [11]). These techniques had not been applied to
x-ray data, however, due to the difficulties associated with simultaneously measuring
a large number of high resolution spectra, each with its own view through the plasma.
Even a relatively modest tomographic scheme requires of order 10 lines of sight. If
a separate spectrometer is required for each line of sight, then these requirements
become extremely onerous in port space, funds and labor.
In their 1999 paper, B.S. Fraenkel and M. Bitter [12] proposed a novel solution to
this problem. By using a spherically bent crystal and a 2d x-ray detector arranged


in the Johann configuration [13], it is possible to obtain multiple lines of sight from a
single spectrometer. Early attempts to implement such a solution were frustrated by
the lack of a large area x-ray detector capable of handling the high count rates necessary for useful measurements of rapidly evolving plasmas [14]. Recently, however,
just such a detector was developed at the PSI in Switzerland. It is this dramatic improvement in x-ray detector technology that has made high resolution imaging x-ray
spectroscopy of fusion plasmas practical.
This chapter will describe various concepts and considerations essential to the
design of a high resolution image x-ray crystal spectrometer.

2.1

Bragg Reflection

X-ray crystal spectrometers are based on the principle that the regular spacing of
atoms in a crystal lattice can be used as a diffraction grating. Bragg reflection takes
place when the distance traveled by photons reflecting off adjacent layers of a crystal
is an integral multiple of the photon wavelength.

This gives rise to constructive

interference of the reflected waves as shown figure 2-1. From this diagram one can
readily derive the famous Bragg condition:

nA = 2d sin Ob

(2.1)

This expression relates the resonant angle, Ob, to the wavelength of the photons,
A, the inter-atomic spacing of the crystal, d, and the order of reflection, n. The interatomic spacing of crystals is typically a few angstroms[15]. Figure 2-2 shows how a
flat crystal, a slit and a position sensitive x-ray detector can be arranged to form a
spectrometer (angles have been exaggerated for illustrative purposes).
Note that all the photons reaching a given point on the detector have the same
Bragg angle. Since there is a one-to-one correspondence between 0 and A via the
Bragg condition, it follows that the intensity measured at a point on the detector is
from photons of a specific wavelength.